Solving optimal control problems with non-smooth solutions using an integrated residual method and flexible mesh

TL;DR

This paper introduces an integrated residual method with a flexible mesh for solving optimal control problems with discontinuous solutions, improving accuracy and convergence over traditional fixed mesh approaches.

Contribution

The method allows for capturing solution discontinuities by optimizing time-mesh nodes as decision variables and employs a sequence of least-squares problems to ensure dynamic accuracy.

Findings

Successfully captures discontinuities in solutions

Achieves superlinear convergence with increased mesh intervals

Handles problems with chattering solutions effectively

Abstract

Solutions to optimal control problems can be discontinuous, even if all the functionals defining the problem are smooth. This can cause difficulties when numerically computing solutions to these problems. While conventional numerical methods assume state and input trajectories are continuous and differentiable or smooth, our method is able to capture discontinuities in the solution by introducing time-mesh nodes as decision variables. This allows one to obtain a higher accuracy solution for the same number of mesh nodes compared to a fixed time-mesh approach. Furthermore, we propose to first solve a sequence of suitably-defined least-squares problems to ensure that the error in the dynamic equation is below a given tolerance. The cost functional is then minimized subject to an inequality constraint on the dynamic equation residual. We demonstrate our implementation on an optimal control problem that has a chattering solution. Solving such a problem is difficult, since the solution involves infinitely many switches of decreasing duration. This simulation shows how the flexible mesh is able to capture discontinuities present in the solution and achieve superlinear convergence as the number of mesh intervals is increased.